Analysis and algebra on differentiable manifolds

After recalling some definitions and results on the basics of smooth manifolds, this chapter is devoted to solve problems including (but not limited to) the following topics: Smooth mappings, critical points and critical values, immersions, submersions and quotient manifolds, construction of manifolds by inverse image, tangent bundles and vector fields, with integral curves and flows. Functions and other objects are assumed to be of class C∞ (also referred to either as ‘differentiable’ or ‘smooth’), essentially for the sake of simplicity. Similarly, manifolds are assumed to be Hausdorff and second countable, though we have included a section that analyses what happens when one of these properties fails to hold. We thus try to elicit in the reader a better understanding of the meaning and importance of such properties. On purpose, we have sprinkled this first chapter with many examples and figures. As an instructive example, we prove in detail that the manifold of affine straight lines of the plane, the 2-dimensional real projective space RP2 minus a point, and the infinite Mobius strip are diffeomorphic. As important and non-trivial examples of differentiable manifolds, the real projective space RP and the real Grassmannian Gk(R) are studied in detail. Lines of latitude and longitude began crisscrossing our worldview in ancient times, at least three centuries before the birth of Christ. By A. D. 150, the cartographer Ptolemy had plotted them on the twenty-seven maps of his first world atlas. DAVA SOBEL, Longitude, Walker & Company, New York, 2007, pp. 2–3. (With kind permission from the author and from Walker & Company publishers.) A differentiable manifold is generally defined in one of two ways; as a point set with neighborhoods homeomorphic with Euclidean space En, coordinates in overlapping neighborhoods being related by a differentiable transformation (. . . ) or as a subset of En, defined near each point by expressing some of the coordinates in terms of the others by differentiable functions (. . . ). The first fundamental theorem is that the first definition is no more general than the second (. . . ) P.M. Gadea et al., Analysis and Algebra on Differentiable Manifolds, Problem Books in Mathematics, DOI 10.1007/978-94-007-5952-7_1, © Springer-Verlag London 2013 1 2 1 Differentiable Manifolds HASSLER WHITNEY, “Differentiable Manifolds,” Ann. of Math. 37 (1936), no. 3, p. 645. (With kind permission from the Annals of Mathematics.) 1.1 Some Definitions and Theorems on Differentiable Manifolds Definitions 1.1 Let M be a topological space. A covering of M is a collection of open subsets of M whose union is M . A covering {Uα}α∈A of M is said to be locally finite if each p ∈M has a neighbourhood (an open subset of M containing p) which intersects only finitely many of the sets Uα . A Hausdorff space M is called paracompact if for each covering {Uα}α∈A of M there exists a locally finite covering {Vβ}β∈B which is a refinement of {Uα}α∈A (that is, each Vβ is contained in some Uα). It is known that a locally compact Hausdorff space which has a countable base is paracompact. A locally Euclidean space is a topological space M such that each point has a neighbourhood homeomorphic to an open subset of the Euclidean space R. In particular, such a space is locally compact and paracompact. If φ is a homeomorphism of a connected open subset U ⊂M onto an open subset of R, then U is called a coordinate neighbourhood; φ is called a coordinate map; the functions x = t i ◦ φ, where t i denotes the ith canonical coordinate function on R, are called the coordinate functions; and the pair (U,φ) (or the set (U,x1, . . . , x)) is called a coordinate system or a (local) chart. An atlas A of class C∞ on a locally Euclidean space M is a collection of coordinate systems {(Uα,φα) : α ∈A} satisfying the following two properties: (i) ⋃ α∈AUα =M ; (ii) φα ◦ φ−1 β is C∞ on φβ(Uα ∩Uβ) for all α,β ∈A. A differentiable structure (or maximal atlas) F on a locally Euclidean space M is an atlas A = {(Uα,φα) : α ∈ A} of class C∞, satisfying the above two properties (i) and (ii) and moreover the condition: (iii) The collection F is maximal with respect to (ii), that is, if (U,φ) is a coordinate system such that φ ◦ φ−1 α and φα ◦ φ−1 are C∞ on φα(U ∩ Uα) and φ(U ∩Uα), respectively, then (U,φ) ∈F . A topological manifold of dimension n is a Hausdorff, second countable, locally Euclidean space of dimension n. A differentiable manifold of class C∞ of dimension n (or simply differentiable manifold of dimension n, or C∞ manifold, or smooth nmanifold) is a pair (M,F ) consisting of a topological manifold M of dimension n, together with a differentiable structure F of class C∞ on M . The differentiable manifold (M,F ) is usually denoted by M , with the understanding that when one speaks of “the differentiable manifold” M one is considering the locally Euclidean space M with some given differentiable structure F . Let M and N be differentiable manifolds, of respective dimensions m and n. A map Φ : M → N is said to be C∞ provided that for every coordinate system (U,φ) on M and (V ,ψ) on N , the composite map ψ ◦Φ ◦ φ−1 is C∞. 1.1 Some Definitions and Theorems on Differentiable Manifolds 3 A diffeomorphism Φ : M →N is a bijective C∞ map such that the inverse map Φ−1 is also C∞. The tangent space TpM to M at p ∈ M is the space of real derivations of the local algebra C∞ p M of germs of C∞ functions at p, i.e. the R-linear functions X : C∞ p M →R such that X(fg)= (Xf )g(p)+ f (p)Xg, f,g ∈ C∞ p M. Let C∞M denote the algebra of differentiable functions of class C∞ on M . The differential map at p of the C∞ map Φ : M →N is the map Φ∗p : TpM → TΦ(p)N, (Φ∗pX)(f )=X(f ◦Φ), f ∈ C∞N. Theorem 1.2 (Partition of Unity) Let M be a manifold and let {Uα}α∈A be a locally finite covering of M . Assume that each closure Uα is compact. Then there exists a system {ψα}α∈A of differentiable functions on M such that (a) Each ψα has compact support contained in Uα ; (b) ψα 0 and ∑ α∈Aψα = 1. Definition 1.3 A parametrisation of a surface S in R3 is a homeomorphism x : U ⊂R2 −→ V ∩ S, where U is an open subset of R2 and V stands for an open subset of R3, such that x∗p : R2 →R3 is injective for all p ∈U . Remark 1.4 In the present book, we use parametrisations of surfaces from open subsets of R2 (or other R to parametrise n-dimensional smooth manifolds) in order to make them consistent with the fact that the coordinate neighbourhoods defined in Definitions 1.1 above are open subsets of the relevant surface (or smooth n-manifold). Moreover, for the sake of simplicity, we usually give only a parametrisation, although it is necessary almost always to give more parametrisations to cover the surface (or other spaces). This should be understood in each case. Definitions 1.5 The stereographic projection from the north pole N = (0, . . . ,0,1) (resp., south pole S = (0, . . . ,0,−1)) of the sphere S = { ( x1, . . . , xn+1 ) ∈Rn+1 : n+1 ∑